Book Reviews
Bednarz, N.; Kieran, C.; Lee, L. (Eds.):
Approaches to Algebra
Perspectives for Research and Teaching
Dordrecht: Kluwer, 1996. – 360 p.
(Mathematics Education Library; 18)
ISBN 0-7923-4145-7
Kaye Stacey, Mollie MacGregor, Melbourne
What should be the main ideas of school algebra? What
approaches might be used in the classroom to develop
them? How might computer technology enhance algebra
learning or change its focus? If you are concerned with
any of these questions, read this book. The sixteen contributing authors present a diversity of views, setting the
stage for spirited debate on the goals and content of the
school algebra curriculum. As the title of the book indicates, a major concern is which of several approaches is
appropriate for beginners.
The book is divided into six sections, most sections comprising several chapters. Four aspects of algebra, each providing a different starting point or initial emphasis for an
algebra course, make up the central topic of the book.
They are:
s algebra as the expression of generality;
s algebra as a problem-solving tool;
s algebra as modeling, using a variety of representations;
s algebra as the study of functions.
Each of these aspects emphasises a different way of giving
meaning to elementary algebra and provides the theme for
a section of the book. Within each section, authors of
individual chapters describe their own opinions, theories
or research findings, or reflect on what others have written
in previous chapters. The reflective chapters are useful
features of this book.
As a prelude to the sections that look at algebra from
the four perspectives, there is a valuable section on the
history of algebra, describing its roots in arithmetic, geometry, and problem-solving and commenting on the status
of “the unknown” in early mathematics. The three chapters in this section, by Charbonneau, Radford and Rojano
respectively, alert us to the very slow development of algebra over centuries and the intellectual effort involved.
These considerations provide a backdrop for the design
of an algebra curriculum. Knowledge of the history of
the growth of mathematical concepts might help us understand how students construct them today. However, as
Wheeler warns, history may not be a trustworthy guide
to the construction of an algebra curriculum for today’s
students.
Charbonneau explains through examples how algebra as
a “science of relations” arose from geometric reasoning.
He also shows the origins of the Cartesian method of analysis – the process of assuming the unknown as known in
order to find it. Radford gives historical examples of the
origins of algebra in both arithmetic and geometry and
traces the development of the concepts of unknown and
variable. He believes that history gives a better understanding of the significance of knowledge rather than prescribing any approach to teaching. Rojano contends that
the separation that students make between algebraic manipulation and the use of algebra in problem-solving may
originate in an oversimplified approach to teaching which
hides the significance of its origins and the semantic background of its grammar. She provides historical examples
to illustrate her observations.
The book presents a range of views on what algebra is
and what it means to “think algebraically”. Some points of
view and suggested teaching strategies reflect an author’s
personal beliefs about mathematics in general and algebra
in particular, whereas others are based on the results of research studies or on practical experience in the classroom.
Mason’s view, for example, is that algebraic thinking –
noticing sameness and difference, seeing generality in patterns, classifying and labelling – is the essential foundation of algebraic behaviour. Through a series of particular
examples, Mason develops the point that expressing generality is the key process underlying deep understanding
of algebra. Teaching experiments described by Lee indicate that generalising activities are a sound introduction
to algebraic thinking and can generate enthusiasm and excitement even in weak students. Lee presents algebra as a
culture; students have to learn to function comfortably in
the new algebraic culture whilst maintaining links to the
old arithmetic culture. She claims that the modern practice of staying within the old culture (arithmetic) while
learning to operate in the new one (algebra) is a cause of
the difficulties and obstacles enumerated in the literature.
As Radford explains, algebraic thinking involves not
only the generalising that leads to concepts of variable
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Book Reviews
and function but also the analytic reasoning that is involved in problem-solving. Analytic reasoning in algebra
requires using unknowns as abstract objects that can be
manipulated, in contrast with arithmetic methods where
reasoning begins with what is known. Many authors are
concerned with how to move students from one way of
reasoning to the other. Have we any reason to assume
that the transition must evolve smoothly? The path may
be discontinuous and non-linear. Most contributors to this
book, however, seem to believe that the transition can be
smooth, via generalising geometric patterns, constructing
spreadsheets, developing formulas for solving problems,
or seeing graphical representations of changing values of
a variable. In fact we see here a proliferation of alternatives to the currently established method of introducing
algebra as primarily a rule-based game of transforming expressions. How was it that this approach, now universally
condemned, became established as the norm? A review of
the history of the teaching of algebra would make interesting reading.
If school algebra begins as a way to solve certain problems, what should these problems be like? Bell proposes
some generic problems as authentic algebraic activities
that should support all aspects of algebra learning. His
method is to begin with a prototype problem in which
students themselves can change the elements and structure, thereby exploring ways of dealing with a general
class of problem situations. Bednarz and Janvier compare
the complexity of typical word problems. They identify
features of the problems that make them amenable to either arithmetic or algebraic thinking. The results of their
empirical study reveal the contrast between the representation and management of quantities and relationships in
spontaneous arithmetic thinking and the kind of reasoning required for algebra. These authors question the value
of the currently popular numeric trials approach, which
encourages students to initially solve problems by trial
and error. Because this procedure encourages children to
think of successive calculations rather than general relationships, it does not lead on to the algebraic solution. On
the other hand, Rojano provides evidence that numeric
trials on a spreadsheet did help some students solve “realworld” problems and build a bridge from thinking about
specific quantities to recognising a general pattern or relationship. She presents some interview evidence that there
are substantial differences between the concept of variation of the unknown developed by the spreadsheet method
and the concept of specific unknown underlying informal
trial and error.
The possibilities offered by new technology feature
strongly in the book, both as an aid to developing traditional algebra and as a doorway to new curriculum
goals. Computing technology provides access to representations of variables as quantities with changing values.
Nemirovsky describes how students are able to relate the
motion of a toy car with the graphs of its speed or distance
by using a motion detector linked to a computer. Heid then
asks whether the facility to represent motion by graphing will lead to a new kind of mathematical knowledge.
lf so, how important is pencil-and-paper symbol manip132
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ulation in the early phases of algebra learning? Both the
modeling approach and the functional approach to algebra are discussed in the context of technology-supported
instruction. Nemirovsky is concerned with students interacting with graphical representations for developing concepts of rates of change. Heid, who also sees computer
modeling as a gateway to algebraic thinking, describes
features of Computer Intensive Algebra which is based
on experiences with functions, and she reports the results
of some classroom experiments. In their courses, students
are provided with function graphers, symbol manipulators
and table generators, enabling them to explore families of
functions. This is the only part of the book that touches
on the looming question of what to do with computer algebra. Kieran, Boileau and Garançon report a series of
investigations with the CARAPACE software that caters
for beginning students who focus on operations rather than
on structures, and builds their algebraic knowledge from
natural-language expressions of algorithms. Later there are
interesting comments on the differences between a pointwise functional approach (which emphasizes that the function turns each input number into an output number in a
regulated way) and a functional approach (which stresses
features of the rate of change of functions).
The need for a more thorough examination of the terms
used in mathematics research and more agreement about
meaning is taken up in Claude Janvier’s useful chapter
on “Modeling and the Initiation into Algebra”. He discusses what is unknown about unknowns, what are the
differences between functions, formulas and equations (although they may look identical), and the usual definition of
modeling. The term “modeling” had been used quite differently by Nemirovsky in an earlier chapter, to describe
the process of interpreting a graph in terms of a situation.
After reading this book the reader is aware of a tension
between the different goals for students’ learning that the
various authors have in mind. At one extreme, a philosophical perspective places emphasis on concepts that inhabit
the mathematical world – the Platonic ideals (e.g., functions) whose shadows (e.g., graphs) are seen on the walls
of a cave. At the other extreme, a pragmatic approach
tries to find out what makes sense to students (e.g., numeric trials to solve a problem), emphasising a familiarity
with what is on the floor of the cave first before an investigation of the shadows on the walls and their source. The
computer-enriched approach can be seen as moving students to a new cave equipped with the latest technology.
With motivation and support from the multiple representation systems that technology provides, it is hoped that
students will see beyond the shadows and catch frequent
glimpses of the glorious mathematical universe.
Much research in the classroom is needed involving
practising teachers and ordinary students before anyone
can say where the various approaches would lead if they
were widely adopted. There is consensus at present about
the goals of algebra learning for bright and highly motivated students. However it is hard to decide on a first
priority for weaker students – what we want them to be
able to do with algebra and why they are learning it. Is
there a minimum level of algebra understanding that all
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students should reach, and what would it be about: notation, functions, generalisation, or analytic problem solving? In his reflections at the end of the book, Wheeler asks
whether mathematics educators distinguish sufficiently between what learning is important for society as a whole
and what is important for each individual. All students
should have the chance to enter what Lee calls “the algebra culture”, but how far should each be expected to
go?
The chapters of the book concerned with current approaches to teaching offer a variety of opinions and some
degree of consensus about certain curricular decisions.
Wheeler’s commentary in the final chapter does not attempt to draw together the many issues discussed but summarises their differences and the important questions they
raise. Like many edited books on aspects of mathematics education, this one does not attempt to compare, contrast and synthesise information provided in the various
chapters. Instead, it leaves this task to the reader. As one
example, consider the use of numeric trials for solving
word problems as opposed to an algebraic method. This
is touched on in the chapters by Bednarz and Janvier, in
Rojano’s chapter on spreadsheet approaches, and in the
chapter by Kieran, Boileau and Garançon. All these authors agree that there is an important difference between
arithmetic and algebraic solution patterns. Bednarz and
Janvier suggest that the gulf between numeric trials and
algebra is greater than the gulf between logical arithmetic
reasoning and algebra. They believe this is due to the fact
that the use of numeric trials never requires integrating relationships in the problem but always works on one relationship at a time. However Kieran, Boileau and Garançon
refer to the effectiveness of the CARAPACE software in
building a bridge between numeric trials and algebra. In
this approach, numeric trials are used in such a way that
they develop understanding of relationships instead of being merely a means for getting answers. Rojano takes a
different view, seeing numeric trials as “a real foundation
upon which the methods or strategies of algebraic thought
are constructed”. In Rojano’s program, students set up a
spreadsheet with an initial cell for the unknown, and use
this cell as a reference for expressing the rest of the unknowns and data. The value of the unknown is then varied
until the problem conditions are met, giving the answer to
the problem. Together the three chapters provide insights
into the different ways numeric trials can be used and
how they relate to algebraic thinking. However the three
groups of authors do not refer to each others’ work, and
the editors have not attempted to draw it together.
The book is to be read as a whole for the arguments
it presents and the questions it provokes. Since it focuses
on the work of the contributors, it is clearly not intended
to provide an overview of research in the field of initial
school algebra instruction. The chapter called “Technology and a Functional Approach to Algebra”, for example,
contains only a few references to related studies. Many
readers, especially those new to the literature on algebra
learning, may be puzzled by some of the terms used. A
dictionary will not help you find a meaning for “syncopation of algebra”, “the didactic cut”, “affection of a power”
Book Reviews
(is this a translation from another language?) or “epistemic
norms”. However most of the writing is clear and the important messages it conveys will be understood. The book
will be a valuable stimulus for discussion amongst teachers, researchers, and all those interested in the purposes
and content of school algebra courses.
Authors
Stacey, Kaye, Prof. Dr., University of Melbourne, Department
of Science & Mathematics Education, Parkville, Victoria
3052, Australia
MacGregor, Mollie, Dr., University of Melbourne, Department
of Science & Mathematics Education, Parkville, Victoria
3052, Australia. m.macgregor@edfac.unimelb.edu.au
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